WASHINGTON—The White House doctor nominated to run the Veterans Affairs department will continue to seek confirmation despite concerns about his performance and behavior in his current job that were publicly aired Tuesday.

In a private meeting on Tuesday, President Donald Trump asked Dr. Ronny Jackson about the allegations, a White House official said, and Dr. Jackson told him they weren’t true. When Mr. Trump asked how he wanted to handle the matter, Dr. Jackson he wanted to fight the accusations and not drop out, the official said, and Mr. Trump agreed to the approach.

Sen. Jon Tester of Montana, the top Democrat on the Veterans’ Affairs Committee, laid out the allegations in a CNN interview Tuesday night, saying the committee was investigating allegations from active duty and retired military officers who had worked in the White House medical office and who had come to the committee with numerous complaints, including that Dr. Jackson belittled lower-ranking employees, drank to excess on overseas trips, and handed out prescriptions “like candy,” walking down the aisle during flights doling out drugs to help people fall asleep and wake up.

He said that Dr. Jackson allegedly was drunk while accompanying President Barack Obama on overseas trips, so that he “couldn’t respond because he had been drinking so much,” leaving someone else to fill in. Mr. Tester also said that Dr. Jackson’s treatment of workers created an unbearable work environment.

“We were told time and time again the people above him he treated like gold, the people below him he belittled, screamed at,” Mr. Tester said. The result was “a very toxic environment to the point where the people who worked around him felt like they had to walk on eggshells because of his lack of respect for their jobs.”

The White House didn’t make Dr. Jackson available for an interview.

Mr. Tester’s comments added to the uncertainty over Dr. Jackson’s fate. But the White House indicated Tuesday it is fighting back against suggestions that Dr. Jackson should withdraw amid the allegations, releasing written evaluations concluding that the nominee had a spotless record with no hint of impropriety.

“Dr. Jackson’s record as a White House physician is impeccable,” a senior White House official said. “He has improved unit morale, received glowing reviews and promotions under Republican and Democratic presidents, and has been given a clean vet from the FBI.”

Earlier in the day, Mr. Trump said he would sympathize if Dr. Jackson withdrew, telling a news conference: “If I were him…I wouldn’t do it.”

Recounting a conversation with Dr. Jackson, who serves as White House physician, Mr. Trump said he had asked: “What do you need this for?”

Appearing on Capitol Hill for meetings, Dr. Jackson was asked about allegations of drinking on the job, creating a hostile work environment and overprescribing medicine. Could he say they were categorically untrue? he was asked.

Dr. Jackson said: “I’m looking forward to the hearing so I can sit down and explain everything.”

Asked if there was an inspector general report reviewing such allegations, he said: “No, there was not.”

In response to Mr. Tester’s findings, the White House released work evaluations and testimonials portraying the nominee as an exemplary public official.

Former President Barack Obama signed a fitness report in 2016 that included a hand-written note: “Ronny does a great job—genuine enthusiasm, poised under pressure, incredible work ethic and follow through,” Mr. Obama wrote. “… Continue to promote ahead of peers.”

A report signed by Mr. Trump last year carried a handwritten note: “Dr. Jackson is a great doctor + leader – ‘2 star material.’ ”

Dr. Jackson, a U.S. Navy rear admiral, faces a tough confirmation fight in the Senate. Trouble signs emerged Monday night when the Senate veterans’ affairs panel indefinitely postponed a confirmation hearing that had been set for Wednesday.

Senators want more records involving Dr. Jackson before they reschedule the hearing.

Committee Chairman Johnny Isakson (R., Ga.) said Tuesday that he hadn’t received any documents from the White House that he had requested.

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“They haven’t given me anything,” he told reporters. Mr. Isakson said he intended to set a new date for Dr. Jackson’s confirmation hearing, but didn’t know when.

Messrs. Isakson and Tester sent a letter to the White House requesting that the Trump administration turn over any allegations or findings against Dr. Jackson. They asked for records of investigations involving him as the president’s doctor or when he worked in the White House Medical Unit dating back to 2006.

An audit done in October 2012, when Dr. Jackson served as head of the White House Medical Unit under Mr. Obama, described an unhappy workplace marked by clashes between him and another senior colleague.

A copy of the audit, reviewed by The Wall Street Journal, reads: “The staff characterize the working environment as being caught between parents going through a bitter divorce, with one parent undermining and talking bad about the other.”

It also described “a severe and pervasive lack of trust in the leadership that has deteriorated to the point that staff walk on ‘eggshells.’ ”

In his response to the audit, Dr. Jackson said that morale was far better than the report depicted.

The White House released an audit from the following year, 2013, that showed healthy majorities of medical unit staff felt “valued” and agreed that the work atmosphere had improved.

After ousting David Shulkin last month, the president nominated Dr. Jackson to run the VA, the second-largest agency in the federal government next to the Pentagon.

The pick raised concerns inside the White House. Chief of Staff John Kelly was wary about nominating Dr. Jackson, uncertain whether he would get a warm reception on Capitol Hill, where he had few relationships, a White House official said Tuesday.

When the nomination was announced, some Republican lawmakers questioned whether the White House doctor had the experience needed to run a 350,000-person bureaucracy enmeshed in repeated scandals over the medical care provided to veterans.

White House officials said Tuesday that Dr. Jackson would need to decide whether he has “the stomach” for what could be a bruising confirmation battle.

As the day wore on, Dr. Jackson assured them that he indeed wanted to persevere.